On Differentiable Vectors for Representations of Infinite Dimensional Lie Groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations \pi \: G \to \GL(V) of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^\infty$ of smooth vectors. If $G$ is a Banach--Lie group, we define a topology on the space $V^\infty$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^\infty$ is a Fr\'echet space. This applies in particular to $C^*$-dynamical systems (\cA,G, \alpha), where $G$ is a Banach--Lie group. For unitary representations we show that a vector $v$ is smooth if the corresponding positive definite function \la \pi(g)v,v\ra is smooth. The second class of results concerns criteria for $C^k$-vectors in terms of operators of the derived representation for a Banach--Lie group $G$ acting on a Banach space $V$. In particular, we provide for each $k \in \N$ examples of continuous unitary representations for which the space of $C^{k+1}$-vectors is trivial and the space of $C^k$-vectors is dense.Comment: 44 pages, Lemma 5.2 and some typos correcte

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Let $G$ be a 1-connected Banach-Lie group or, more generally, a BCH--Lie group. On the complex enveloping algebra U_\C(\g) of its Lie algebra \g we define the concept of an analytic functional and show that every positive analytic functional $\lambda$ is integrable in the sense that it is of the form \lambda(D) = \la \dd\pi(D)v, v\ra for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of *-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. For the matrix coefficient \pi^{v,v}(g) = \la \pi(g)v,v\ra of a vector $v$ in a unitary representation of an analytic Fr\'echet-Lie group $G$ we show that $v$ is an analytic vector if and only if $\pi^{v,v}$ is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a 1-connected Fr\'echet--BCH--Lie group $G$ extends to a global analytic function.Comment: Minor revision